Engineering Geology 168 (2014) 120–128

Contents lists available at ScienceDirect

Engineering Geology

j ourna l homepage: www.e lsev ie r .com/ locate /enggeo

Slope reliability analysis considering spatially variable shear strengthparameters using a non-intrusive stochastic finite element method

Shui-Hua Jiang a, Dian-Qing Li a,⁎, Li-Min Zhang b, Chuang-Bing Zhou a

a State Key Laboratory of Water Resources and Hydropower Engineering Science, Key Laboratory of Rock Mechanics in Hydraulic Structural Engineering (Ministry of Education),Wuhan University, 8 Donghu South Road, Wuhan 430072, PR Chinab Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

⁎ Corresponding author at: State Key Laboratory of WaEngineering Science, Wuhan University, 8 Donghu SoChina. Tel.: +86 27 6877 2496; fax: +86 27 6877 4295.

E-mail address: dianqing@whu.edu.cn (D.-Q. Li).

0013-7952/$ – see front matter © 2013 Elsevier B.V. All rihttp://dx.doi.org/10.1016/j.enggeo.2013.11.006

a b s t r a c t

a r t i c l e i n f o

Article history:Received 31 July 2013Received in revised form 31 October 2013Accepted 10 November 2013Available online 16 November 2013

Keywords:SlopesShear strengthSpatial variabilityRandom fieldReliabilityStochastic finite element method

This paper proposes a non-intrusive stochastic finite element method for slope reliability analysis consideringspatially variable shear strength parameters. The two-dimensional spatial variation in the shear strength param-eters is modeled by cross-correlated non-Gaussian random fields, which are discretized by the Karhunen–Loèveexpansion. The procedure for a non-intrusive stochastic finite element method is presented. Two illustrativeexamples are investigated to demonstrate the capacity and validity of the proposed method. The proposednon-intrusive stochastic finite element method does not require the user to modify existing deterministic finiteelement codes, which provides a practical tool for analyzing slope reliability problems that require complex finiteelement analysis. It can also produce satisfactory results for low failure risk corresponding tomost practical cases.The non-intrusive stochastic finite element method can efficiently evaluate the slope reliability consideringspatially variable shear strength parameters, which is much more efficient than the Latin hypercube sampling(LHS) method. Ignoring spatial variability of shear strength parameters will result in unconservative estimatesof the probability of slope failure if the coefficients of variation of the shear strength parameters exceed a criticalvalue or the factor of slope safety is relatively low. The critical coefficient of variation of shear strength parametersincreases with the factor of slope safety.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

In recent years, the spatial variability of soil properties has receivedconsiderable attention in slope stability analysis. Many investigatorshave contributed to this subject (e.g., Griffiths and Fenton, 2004; Cho,2007; Low et al., 2007; Srivastava and Sivakumar Babu, 2009; Cho,2010; Srivastava et al., 2010; Griffiths et al., 2011; Wang et al., 2011;Cho, 2012; Ji et al., 2012; Li et al., 2013c; Zhu and Zhang, 2013). For ex-ample, Griffiths and Fenton (2004) studied the effect of spatial variabil-ity of undrained shear strength on the probability of slope failure usingrandom finite element method. Cho (2007) investigated the effect ofspatially variable soil properties on the slope stability using directMonte Carlo simulations (MCS). Low et al. (2007) proposed a practicalEXCEL procedure to analyze slope reliability in the presence of spatiallyvarying shear strength parameters. Srivastava and Sivakumar Babu(2009) quantified the spatial variability of soil parameters using fieldtest data and evaluated the reliability of a spatially varying cohesive–frictional soil slope. Cho (2010) investigated the effect of spatial

ter Resources and Hydropoweruth Road, Wuhan 430072, PR

ghts reserved.

variability of shear strength parameters accounting for the correlationbetween cohesion and friction angle on the slope reliability. Srivastavaet al. (2010) investigated the effect of spatial variability of permeabilityparameter on steady state seepage flow and slope stability. Griffithset al. (2011) performed a probabilistic analysis to explore the influenceof spatial variation in the shear strength parameters on the reliability ofinfinite slopes.Wang et al. (2011) developed a subset simulation-basedreliability approach for slope stability analysis considering spatially var-iable undrained shear strength. Ji et al. (2012) adopted the First OrderReliability Method (FORM) coupled with a deterministic slope stabilityanalysis to search the critical slip surface when the spatial variability inthe shear strength parameters is considered.

In the majority of these studies, the traditional limit equilibriummethod (LEM) is used to perform deterministic slope stability analyses.Then, the LEM is combinedwith random field theory for slope reliabilityanalysis considering spatially variable soil properties. Thereafter, MonteCarlo Simulation is used to evaluate the probability of slope failure. Apotential pitfall of the LEM is that some assumptions relating to theshape or location of the critical failure mechanism have to be made.Also, it does not account for the stress–strain behavior of the soil. Addi-tionally, the spatial variability of soil properties cannot be consideredrealistically with the LEM-based methods, unless the shape of the slipsurface is non-circular (Tabarroki et al., 2013). Fortunately, finite elementbased methods provide solutions to overcome the aforementioned

121S.-H. Jiang et al. / Engineering Geology 168 (2014) 120–128

shortcomings underlying the traditional LEM (Farias and Naylor, 1998;Griffiths and Fenton, 2004). As for the slope reliability evaluation, al-though the direct MCS is simple and suitable for evaluating the probabil-ity of slope failure in the presence of spatially variable shear strengthparameters, the time and resources required for theMCS could be prohib-itive because a substantial number offinite elementmodel runs are need-ed to obtain reliability results with a sufficient accuracy. The resultantcomputational efforts aremost pronounced at relatively small probabilitylevels or when complex finite element analyses are needed for slopestability analysis. Traditional stochastic finite element methods requiresignificant modification of existing deterministic numerical codes, andbecomenearly impossible formost engineerswith no access to the sourcecodes of commercial software packages (Ghanem and Spanos, 2003;Stefanou, 2009). Therefore, it is necessary to explore more efficientmethods for slope reliability analysis, which considers spatially variableshear strength parameters and requires complex finite element analysisfor determining the factor of safety.

The objective of this paper is to propose a non-intrusive stochasticfinite elementmethod for slope reliability analysis considering spatiallyvariable shear strength parameters. To achieve this goal, this articleis organized as follows. In Section 2, the two-dimensional (2-D) spatialvariation of the shear strength parameters is modeled by cross-correlated non-Gaussian random fields, which are discretized by theKarhunen–Loève (KL) expansion. In Section 3, the procedure of a non-intrusive stochastic finite element method is presented. Two examplesof slope reliability analysis are investigated to demonstrate the capacityand validity of the proposed method in Section 4.

2. Random field modeling of soil property

2.1. Spatial variability of soil property

A Gaussian random field is completely defined by its mean μ(x),standard deviation σ(x), and autocorrelation function ρ(x1, x2). Theautocorrelation function is an important physical quantity for character-izing the spatial correlation of soil properties (Vanmarcke, 2010). In thisstudy, a squared exponential 2-D autocorrelation function is adoptedwith different autocorrelation distances in the horizontal and verticaldirections as follows:

ρ x1; y1ð Þ; x2; y2ð Þ½ � ¼ exp − x1−x2j jlh

� �2þ y1−y2j j

lv

� �2� �� �ð1Þ

where (x1, y1) and (x2, y2) are the coordinates of two arbitrary points ina 2-D space; and lh and lv are the autocorrelation distances in the hori-zontal and vertical directions, respectively.

2.2. Karhunen–Loève (KL) expansion

Several methods such as the midpoint method (Der Kiureghian andKe, 1988), the local average subdivision (LAS) method (Vanmarcke,2010), the shape function method (Liu et al., 1986) and the KL expan-sion (Phoon et al., 2002) can be used to discretize the random field.Since the KL expansion requires the minimum number of randomvariables for a prescribed level of accuracy, it is employed todiscretize the 2-D anisotropic random fields of shear strength parame-ters. To facilitate the understanding of the proposed non-intrusive sto-chastic finite element method, the KL expansion is introduced brieflyin the following.

A random field H(x, θ) is a collection of random variables associatedwith a continuous index x ∈ Ω p Rn, where Ω is an open set of Rn de-scribing the system geometry and θ∈Θ is the coordinate in the outcomespace. Discretization of a random field using the KL expansion is basedon the spectral decomposition of its autocorrelation function ρ(x1, x2).Generally, the autocorrelation function is bounded, symmetric andpositive definite. Hence, the discretization of a random field is defined

by the eigenvalue problem of the homogenous Fredholm integral equa-tion as follows:

ZΩρ x1; x2ð Þ f i x2ð Þdx2 ¼ λi f i x1ð Þ ð2Þ

where x1 and x2 denote the coordinates of two points; fi(·) and λi are theeigenfunctions and eigenvalues of the 1-D autocorrelation functionρ(x1, x2), respectively. Then, the eigenmodes of the separable multi-dimensional autocorrelation function are calculated by multiplyingwith the eigenmodes obtained from Eq. (2) (e.g., Huang, 2001).

The eigenvalue problem of the Fredholm integral equation in Eq. (2)is often solved numerically due to its complexity. The wavelet-Galerkintechnique is adopted herein to solve the above eigenvalue problem.More details are given by Phoon et al. (2002). The series expansion ofa 2-D random field Hi(x, y) is expressed as

Hi x; yð Þ ¼ μ i þX∞j¼1

σ i

ffiffiffiffiffiλ j

qf j x; yð Þξi; j; x; y∈Ω ð3Þ

where ξi,j is a set of orthogonal random coefficients (uncorrelated ran-dom variables with zero mean and unit variance). The series expansionin Eq. (3), referred to as the KL expansion, provides a second-momentcharacterization in terms of uncorrelated random variables and deter-ministic orthogonal functions. It is known to converge in the meansquare sense for any distribution of Hi(x, y) (e.g., Vořechovský, 2008).For practical implementation, the series is approximated by a finitenumber of terms in Eq. (3):

eHi x; yð Þ ¼ μ i þXnj¼1

σ i

ffiffiffiffiffiλ j

qf j x; yð Þξi; j; x; y∈Ω ð4Þ

where n is the number of KL expansion terms to be retained, whichhighly depends on the desired accuracy and the autocorrelation func-tion of the random field. Small values of the autocorrelation distanceswill lead to a significant increase in the number of the eigenmodes, n.Several studies (Huang, 2001; Laloy et al., 2013) took the ratio of the ex-pected energy, ε, as a measure of the accuracy of the truncated series,which is defined as

ε ¼Z

ΩE eHi x; yð Þ−μ i

� �2dxdy=

ZΩE Hi x; yð Þ−μ ið Þ

2dxdy

¼Xni¼1

λi=X∞i¼1

λi ð5Þ

where the eigenvalues λi are sorted in a descending order. A large ε al-ways indicates a high accuracy of the truncated series.

2.3. Cross-correlated non-Gaussian random fields

In geotechnical engineering practice, very often more than one geo-technical parameter needs to be modeled by random fields. Further-more, the geotechnical engineering literature is replete with cross-correlations between two geotechnical parameters. For example, thetwo curve-fitting parameters underlying load–displacement curve ofpiles are negatively correlated (Li et al., 2013b). The cohesion and fric-tion angle, often used for slope reliability analysis, are considered tobe negatively correlated (e.g., Lumb, 1970; Wolff, 1985; Cho, 2010;Tang et al., 2013). In this case, the cross-correlated random fields needto be handled. Following Cho (2010), it is assumed that all fields simu-lated over a regionΩ share an identical autocorrelation function overΩ,and the cross-correlation structure between each pair of simulatedfields is simply defined by a cross-correlation coefficient. It can ensurethat the target random fields respect the correlation structure withineach field (Vořechovský, 2008). The rationale underlying these assump-tions has been explained by Fenton and Griffiths (2003). Under these

Fig. 1. Flowchart of the non-intrusive stochastic finite element method.

122 S.-H. Jiang et al. / Engineering Geology 168 (2014) 120–128

assumptions, the modal decomposition of the given autocorrelationfunction is done only once. The same spectrumof eigenfunctions and ei-genvalues can be used for the expansion of cross-correlated randomfields. It should be pointed out that the sets of random variables usedfor the expansion of cross-correlated random fields are also cross-correlated. In the following, the cross-correlated random fields betweencohesion c and friction angleϕ are presented to illustrate the simulationprocedures of cross-correlated random fields.

Denote the cross-correlation coefficientmatrix between c andϕ asRc,

ϕ = (ρc,ϕ)2 × 2. The samplematrix ξwith the dimension (n × NF) × Np isgenerated first, where Np is the number of simulated samples, or thenumber of deterministic slope stability model runs; NF is the number ofrandom fields. Each of the Np columns is one realization of an indepen-dent standard normal sample vector, which is partitioned into NF

vectors each with the dimension n. For the discretization of cross-correlated random fields involving two spatially variable c and ϕ, thekth column of ξ, ξk = {ξck, ξϕk }, ξck = {ξc,1k ,ξc,2k , ⋯,ξc,nk}T, ξϕk = {ξϕ,1k ,ξϕ,2k , ⋯,ξϕ,nk }T, can be performed using two sets of independent standardnormal probabilistic collocation points (e.g., Li and Zhang, 2007; Liet al., 2013a) or Latin hypercube sampling points (Choi et al., 2004;Vořechovský, 2008). A lower triangular matrix L with a dimension of2 × 2 is obtained by the Cholesky decomposition of Rc,ϕ. Then, the cor-related standard normal samplematrixχ is obtained. The kth columnofχ, χk, is given by

χk ¼ χkc ;χ

kϕ

h i¼ ξk � LT ¼ ξkc ; ξ

kc � ρc;ϕ þ ξkϕ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−ρ2

c;ϕ

qh i; k

¼ 1;2; ⋯;Np ð6Þ

where the matrix ξk with a dimension of n × 2 is rearranged from thekth column of the sample matrix ξ. Taking the correlated standard nor-mal sample vector χk as basis, the kth realization of each of the cross-correlated Gaussian random fields of c and ϕ is expressed as

eHik;D ¼ μ i þ

Xnj¼1

σ i

ffiffiffiffiffiλ j

qf j x; yð Þχk

i; j for i ¼ c;ϕð Þ: ð7Þ

Like the isoprobabilistic transformation of non-normal random vari-ables (e.g., Li et al., 2011), the kth realization of approximate cross-correlated non-Gaussian random fields can be obtained component-to-component,

Hik;NG x; yð Þ ¼ G−1

i Φ eHik;D

x; yð Þh in o

for i ¼ c;ϕð Þ ð8Þ

where Gi−1(·) is the inverse function of marginal cumulative distri-

bution of each component of non-Gaussian vector random fieldHNG(x, y); and Φ(·) is the standard Gaussian distribution function. If theshear strength parameters (c,ϕ) are considered to be lognormally distrib-uted, the kth realization of approximate cross-correlated lognormalrandom fields can be obtained by exponentiating that of the approximatecross-correlated Gaussian random fields from Eq. (8) as below:

Hik;LN x; yð Þ ¼ exp μ ln i þ

Xnj¼1

σ ln i

ffiffiffiffiffiλ j

qf j x; yð Þχk

i; j

0@

1A for i ¼ c;ϕð Þ ð9Þ

where μlni and σlni are the mean and standard deviation of Gaussian

random variable lni, respectively, μlni = ln μi − σln i2 /2 and σ ln i

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiln 1þ σ i=μ ið Þ2 q

.

3. Procedure of a non-intrusive stochastic finite element method

A procedure of slope reliability analysis using a non-intrusivestochastic finite element method is proposed in this section, as shownin Fig. 1. This procedure consists of nine steps as follows:

(1) Identify the spatially varying variables and determine their sta-tistics such as means, coefficients of variation (COVs), distribu-tions and cross-correlation coefficients among the variablesassociated with the slope reliability problem. Select an appropri-ate autocorrelation function and estimate the autocorrelationdistances in the horizontal and vertical directions for a 2-Drandom field model.

(2) Construct the finite element analysis (FEA) model for slope sta-bility analysis with the mean values of input variables usingSIGMA/W and SLOPE/W (GEO-SLOPE International Ltd., 2010a,b). Then, extract the coordinates (xo,i, yo,i) of the centroid of theith element, in which i = 1, 2,⋯, ne, ne is the number of finiteelements. Save the deterministic slope stability model file as aninput file named “FEM-FS.xml”. This file contains all the informa-tion needed by SIGMA/W and SLOPE/W, which can also bedirectly viewed via the text editor. Note that the factor of slopesafety is calculated using the finite element based method (Fariasand Naylor, 1998), which is implemented in SLOPE/W using thestress field obtained from finite element analysis in SIGMA/W.

(3) Generate the independent standard normal sample matrix ξ ofdimension (n × NF) × Np using the probabilistic collocation pointsor Latin hypercube sampling points. Then, transform the indepen-dent standard normal sample matrix ξ into the correlated stan-dard normal sample matrix χ using Eq. (6).

123S.-H. Jiang et al. / Engineering Geology 168 (2014) 120–128

(4) Simulate the cross-correlated non-Gaussian random fields ofspatially varying variables with the sample matrix χ and coordi-nates (xo,i, yo,i) using the KL expansion in Section 2. Then, Np real-izations of cross-correlated non-Gaussian random fields ofspatially varying shear strength values in the physical space areobtained, which are assigned to each finite element of the consid-ered slope, respectively.

(5) Replace the mean values of the corresponding uncertain inputparameters at the centroid of each finite element in the “FEM-FS.xml” file generated in step (2) with each pair of spatially vary-ing variables (i.e., c and ϕ) in each realization of random fields instep (4). Then, Np different new “FEM-FS.xml” input files aregenerated. In this manner, no programming effort is required tomodify the existing finite element code compared with the spec-tral stochasticfinite elementmethod (Ghanemand Spanos, 2003).

(6) Run SIGMA/W and SLOPE/W with each new “FEM-FS.xml” inputfile generated in step (5) to performdeterministic FEAof slope sta-bility. Such a process can be executed automatically with the helpof Winbatch™. Winbatch™ is a Microsoft Windows scriptinglanguage and possesses an optional compiler used to create self-contained executables. This process will produce Np different fac-tors of slope safety, FS = (FS1, FS2,…, FSNp), which can be directlyextracted from the corresponding result files named “FEM-FS.fac”.

(7) Replace the implicit function between the factor of slope safety, FS,and the uncertain input parameters by a Hermite polynomialchaos expansion (PCE) called meta-model when the FEA of slopestability is involved (Isukapalli et al., 1998; Ghanem and Spanos,2003). The PCEmethodology has beenwidely used in geotechnicalengineering (Li et al., 2011; Mollon et al., 2011; Al-Bittar andSoubra, 2013).

FS ξð Þ ¼ a0Γ0 þXNi1¼1

ai1Γ1 ξi1� �

þXNi1¼1

Xi1i2¼1

ai1 i2Γ2 ξi1 ; ξi2� �

þXNi1¼1

Xi1i2¼1

Xi2i3¼1

ai1 i2 i3Γ3 ξi1 ; ξi2 ; ξi3� �

þ � � � þXNi1¼1

Xi1i2¼1

Xi2i3¼1

� � �XiN−1

iN¼1

ai1 i2 ;���; iNΓN ξi1 ; ξi2 ; � � �; ξiN

� �ð10Þ

where N is the total number of random variables, N = n × NF; a

¼ a0; ai1 ; � � �; ai1 i2 ;���; iN� �

are the unknown coefficients to be evaluated; ξi

¼ ξi1 ; ξi2 ; � � �; ξiN� �

is the vector of independent standard normal vari-

ables representing the uncertainties in the input parameters, whichcorresponds to the random variables used to discretize the random fields

using the KL expansion in Eq. (3); andΓN ξi1 ; ξi2 ; � � �; ξiN� �

are themultidi-

mensional Hermite polynomials of degree N. The reader is referred toGhanem and Spanos (2003) and Li et al. (2011) for details. Note thatthe number of unknown coefficients in Eq. (10) is Nc, Nc = (N + p)!/(N! × p!) for the pth order PCE.

(8) Determine the unknown coefficients in the Hermite PCE byequating the factors of safety obtained from step (6) with the es-timates from the series approximation in Eq. (10) at thematrix ξfrom step (3), then constructing and solving a system of linearequations. The collocation method based on the linearly inde-pendent principle (Li and Zhang, 2007) or the regression basedapproach (Isukapalli et al., 1998) can be used for such purpose.Then, the explicit function between FS and the uncertain inputparameters is obtained.

(9) Perform probabilistic analyses on the explicit performance func-tion G(ξ) = FS(ξ) − 1. The probability of failure and the corre-sponding reliability index can be estimated by using the MCSwith one million samples on the performance function with the

FS represented by a Hermite PCE. The first four statistical mo-ments can also be directly evaluated using the PCE (Mollonet al., 2011). It should be pointed out that the evaluation of theperformance function does not require deterministic FEA of slopestability again, but only involve the evaluation of simple algebraicexpressions, which is much more computationally efficient.

4. Illustrative examples

4.1. Application to a saturated clay slope under undrained conditions(ϕu = 0)

In the first example, an undrained clay slope studied by Griffiths andFenton (2004) and Cho (2010) is investigated. A typical finite elementmodel of the considered slope is shown in Fig. 2. The majority of theelements are squares and the elements adjacent to the slope surfaceare degenerated into triangles. The types of elements are 4-node quad-rilateral elements and 3-node triangular elements. In Fig. 2, the finiteelement mesh consists of 910 elements and 981 nodes. For illustrativepurposes, a conventional elastic and perfectly plastic model based onthe Mohr–Coulomb failure criterion is adopted to represent thestress–strain behavior of the soil. The boundary conditions are rollerson both lateral boundaries and full fixity at the base.

To avoid negative values, the undrained shear strength, cu, is consid-ered as a log-normally distributed random field. Table 1 summarizes thestatistical properties of soil parameters for the considered slope. Notethat Young's modulus E, Poisson's ratio v, and unit weight γsat of thesoil are treated as deterministic quantities because their variations arerelatively low compared with cu (Duncan, 2000). Based on the meanvalue of theundrained shear strength, theminimum factor of slope safe-ty is obtained as 1.366 using the finite element based method based onthe search algorithm for the critical failure surface, which is implement-ed in SIGMA/W and SLOPE/W. The corresponding critical failure surfaceis plotted in Fig. 2, which is deep and passes through the foundation soil.For comparison, the factor of safety using the limit equilibriummethod is also calculated for this slope model. The FS is 1.354 usingthe Morgenstern–Price method, which is consistent with 1.356 usingBishop's simplified method reported in Cho (2010). These results indi-cate that the finite element based method adopted in this study evalu-ates the slope stability problem effectively.

For computational efficiency, the KL expansion is employed todiscretize the 2-D log-normally distributed random field of cu. The accu-racy of discretization of random field highly depends on the number ofeigenmodes, n. Generally, an increase in the number of eigenmodes in-creases the accuracy. However, it also increases the computational ef-fort. In practice, a compromise between accuracy and computationalcost is achieved by accepting some amount of error. The wavelet-Galerkin technique is used herein to solve the eigenvalue problem un-derlying the squared exponential autocorrelation function in Eq. (1).Fig. 3 shows the decaying trends of the eigenvalues obtained by solvingthe integral eigenvalue problem. Note that the eigenvalues decay dras-tically with the number of KL terms. Moreover, the rate of decay in-creases with increasing autocorrelation distance. For n = 10, theratios of the expected energy in Eq. (5) are 92.8%, 97.1% and 98.7% for(lh = 20 m, lv = 2 m), (lh = 25 m, lv = 2.5 m) and (lh = 30 m,lv = 3 m), respectively. When n increases from 10 to 15 associatedwith (lh = 25 m, lv = 2.5 m), the ratio of the expected energy, ε, onlyincreases from 97.1% to 99.3%. The improvement in the ratio of the ex-pected energy is less significant. However, the resulting computationaleffort is increased significantly in comparison with that for n = 10. Aratio of ε ≥ 95% is commonly taken as a criterion for determining thenumber of eigenmodes n (Laloy et al., 2013). Taking the autocorrelationdistances of lh = 25 m and lv = 2.5 m as an example, which are typicalvalues reported in the literature (Phoon and Kulhawy, 1999; El-Ramlyet al., 2003), a value of n = 10 is adopted to achieve a compromisebetween accuracy and efficiency in the following.

Distance (m)

0 5 10 15 20 25 30

Ele

vatio

n (m

)

0

2

4

6

8

10

Fig. 2. FEMmodel of the homogeneous undrained clay slope with FS = 1.366.

60

80

100

120

igen

valu

e

lh=20m, l

v=2m

lh=25m, l

v=2.5m

lh=30m, l

v=3m

124 S.-H. Jiang et al. / Engineering Geology 168 (2014) 120–128

The finite element size is an important parameter which can affectthe accuracy of reliability results when the spatial variability is consid-ered. As suggested by Griffiths and Fenton (2004) and Ching andPhoon (2013), the ratio of the element size to the scale of fluctuationhas to be sufficiently small to ensure that the variance function ap-proaches 1.0. For the autocorrelation distances of lh = 25 m andlv = 2.5 m, the ratio of the element size to the vertical scale of fluctua-tion, Δzδv ¼ 0:5

2:5ffiffiffiπ

p ¼ 0:11, is relatively small for a square finite element ofside length 0.5 m. It can be seen from Vanmarcke (1977) that the vari-ance function Γu(Δz) approaches 1.0 when the spatial autocorrelation ismodeled by the squared exponential model. This also agrees with theobservation in Ching and Phoon (2013) that the element size has tobe smaller than 0.13δv to 0.17δv depending on the stress states and fail-ure curve orientations when the squared exponential autocorrelationfunction is adopted. Although a finermesh can produce better estimatesof factor of slope safety and probability of slope failure, the resultingcomputational time increases dramatically. To strike balance betweenaccuracy and computational cost, the 4-node quadrilateral elementsand 3-node triangular elements with an element size of 0.5 m areadopted in this study. Additionally, the finite element mesh should bechanged along with the autocorrelation distance.

To obtain the random field realizations of spatially variable cu, anindependent standard normal sample matrix ξ with dimensions of10 × 66 is first generated for the 2nd order Hermite PCE using the prob-abilistic collocation points (e.g., Li et al., 2011). This sample matrix,ξ10 × 66, and coordinates (xo,i, yo,i), in which i = 1, 2,⋯, 910, are used tosimulate the lognormal random field of cu. Then, a parameter matrixof spatially variable cu with dimensions of 910 × 66 in the physicalspace is obtained from the KL expansion. These 910 values of cu ineach column of the parameter matrix are assigned to each finite ele-ment according to the order of coordinates (xo,i, yo,i). By replacing themean values of cu for each finite element in the original “FEM-FS.xml”file with these 910 values of cu, a new “FEM-FS.xml” input file is gener-ated. Applying the similarmethod, 66 different new “FEM-FS.xml” inputfiles can be obtained. The deterministic FEA of slope stability is carriedout based on these 66 new “FEM-FS.xml” input files via SIGMA/W andSLOPE/W. This process will produce 66 different factors of slope safety,FS = (FS1, FS2,…, FS66), which are extracted from the corresponding re-sult files named “FEM-FS.fac”, respectively. The explicit function be-tween the factor of slope safety and independent standard normalvariables is constructed using the 2nd order Hermite PCE. A system of

Table 1Statistical properties of soil parameters for an undrained clay slope.

Parameter Mean COV

Undrained shear strength cu (kPa) 23 0.3Young's modulus E (MPa) 100 –

Poisson's ratio ν 0.3 –

Unit weight γsat (kN/m3) 20 –

Note: The symbol “–” denotes that the parameter is constant.

linear equations is constructed by equating the factors of safety obtain-ed from the 66 runs of deterministic finite element slope stability anal-ysis with the estimates from the series approximation in Eq. (10). Theunknown coefficients in the 2nd order Hermite PCE can be determinedusing the collocation point method based on the linearly independentprinciple (e.g., Li and Zhang, 2007; Li et al., 2013a). Finally, the probabil-ity of slope failure is obtained as 11.8% using the direct MCS with 106

samples. Similarly, the probability of slope failure is obtained as11.56% from the 3rd order Hermite PCE where the independent stan-dard normal sample matrix ξ with dimensions of 10 × 286 is used.Note that the direct MCS are carried out on the explicit performancefunction represented by the Hermite PCE. The calculation of FS doesnot involve finite elementmodel runs, but only the evaluation of simplealgebraic expressions, which is much more computationally efficient.

Table 2 shows the reliability results of an undrained clay slope forautocorrelation distances of lh = 25 m, lv = 2.5 m using the proposednon-intrusive stochastic finite element method. To validate the pro-posedmethod, the results obtained from the LHSwith 1000 simulationsare also provided in Table 2. The coefficient of variation of the probabil-ity of failure, pf, associatedwith the LHS,COVp f ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−p fð Þ= 1000 � p fð Þp

(e.g., Zhang et al., 2013), is 8.9% and 6.8% for lh = 25 m, lv = 2.5 m(pf = 11.2%) and lh = 1000 m, lv = 1000 m (pf = 17.6%), respective-ly. They are smaller than the commonly used value 10% in the literature.Thus, the results obtained from the LHS with 1000 simulations can betaken as “exact” solutions for this example. The probabilities of failureobtained from the 2nd and 3rd order PCEs are 11.8% and 11.56%, respec-tively. The corresponding relative errors in the probability of failure are5.4% and 3.2% in comparisonwith 11.2% obtained from the LHSmethod.These results indicate that the proposedmethod can produce sufficient-ly accurate probability of failure. Additionally, the numbers of finite ele-ment model runs, Np, for the 2nd and 3rd order PCEs are 66 and 286

0 10 20 30 40 500

20

40

E

Number of KL terms

Fig. 3. Eigenvalues of the autocorrelation function.

Table 2Reliability results of an undrained clay slope for lh = 25 m and lv = 2.5 m.

Method μFS σFS δFS κFS pf (%) β

2nd order PCE (Np = 66) 1.272 0.243 0.457 3.520 11.80 1.1853rd order PCE (Np = 286) 1.312 0.303 0.814 5.501 11.56 1.197LHS (Np = 1000) 1.294 0.252 0.586 3.568 11.20 1.216

Table 3Reliability results of an undrained clay slope for lh = 1000 m and lv = 1000 m.

Method μFS σFS δFS κFS pf (%) β

2nd order PCE (Np = 66) 1.372 0.411 0.834 3.933 18.52 0.8963rd order PCE (Np = 286) 1.371 0.411 0.921 4.490 17.65 0.929LHS (Np = 1000) 1.372 0.412 0.916 4.430 17.60 0.931SRV 1.366 0.414 – – 17.95 0.917

Note: SRV denotes the single random variable approach.

125S.-H. Jiang et al. / Engineering Geology 168 (2014) 120–128

based on the rank of the informationmatrix, respectively, which are justequal to the number of unknown coefficients in the Hermite PCE, Nc. Incontrast, the LHS needs 1000finite elementmodel runs to achieve a rea-sonable accuracy. It is evident that the non-intrusive stochastic finite el-ement method is much more efficient than the LHS method. The firstfour statistical moments of the factor of safety are also listed inTable 2. The computed skewness and kurtosis using the LHS with1000 simulations are 0.586 and 3.568, respectively. These values are 0and 3 for a normal distribution, which suggests that the factor of safetyof the undrained clay slope as a random variable does not follow a nor-mal distribution.

Similarly, Table 3 shows the reliability results of an undrained clayslope for autocorrelation distances of lh = 1000 m and lv = 1000 m.The probabilities of failure obtained from the 2nd and 3rd order PCEsare 18.52% and 17.65%, respectively. Taking 17.6% obtained from theLHSwith 1000 simulations as the “exact” solution, the relative errors as-sociated with the 2nd and 3rd order PCEs are 5.2% and 0.3%, respective-ly. When the undrained shear strength is modeled by a single randomvariable, the probability of slope failure is 17.95% using the approachpresented in Griffiths and Fenton (2004), which is almost identical tothat obtained from the randomfieldmodelwith the autocorrelation dis-tances of lh = 1000 m and lv = 1000 m. Such result clearly indicatesthat when only the probability of failure is of great interest, the randomfield model with the autocorrelation distances of lh = 1000 m andlv = 1000 m is almost equivalent to the single random variablemodel. With regard to the computational efficiency, the numbers of fi-nite element model runs for the 2nd and 3rd order PCEs are 66 and286, respectively. The computational cost for the 2nd order PCE is only

0.0 0.2 0.4 0.6 0.80.0

0.2

0.4

0.6

0.8

1.0

0.70

0.10

0.26

0.43

Pro

babi

lity

of fa

ilure

COV of undrained shear strength

0.57

Fig. 4. Probabilities of failure for various COVs of u

about one quarter of that for the 3rd order PCE. Hence, the non-intrusive stochastic finite element method with a 2nd order PCE isused in the subsequent analyses.

Fig. 4 summarizes the variations of the probability of slope failurewith the coefficient of variation of cu (COVcu) for various factors of slopesafety. The autocorrelation distances of lh = 1000 m and lv = 1000 mrepresent the case of ignoring spatial variation in cu. It can be observedthat ignoring spatial variation will lead to unconservative estimate ofthe probability of slope failure if FS is below 1.0. This result contradictswith thefindings of other investigatorswhoused classical slope reliabilityanalysis tools (e.g., Cho, 2007; Cho, 2010; Griffiths et al., 2011). With theincrease of FS, there are crossover points between the curves associatedwith (lh = 25 m, lv = 2.5 m) and (lh = 1000 m, lv = 1000 m), whichgive a critical value of COVcu. When the COVcu exceeds the criticalvalue, ignoring spatial variation in cu will underestimate the probabilityof slope failure. Suchfindings are consistentwith the observations report-ed in Griffiths and Fenton (2004). Additionally, the critical value of COVcuincreases with increasing FS. An important observation highlighted inFig. 4 is that a FS = 1.3 for the considered slopewould lead to a probabil-ity of failure as high as pf = 18% associated with COVcu = 0.3 and(lh = 25 m, lv = 2.5 m). In geotechnical engineering practice, however,slopes with a factor of safety as high as FS = 1.3 rarely fail. These resultsfurther support the conclusions drawn by Duncan (2000). He stated that“Computing both factor of safety and probability of failure is better thancomputing either one alone. Although neither factor of safety nor proba-bility of failure can be computedwith high precision, both have value andeach enhances the value of the other”.

Another interesting finding observed from Fig. 4 is that the probabil-ity of slope failure associated with FS b 1 does always increases withCOVcu in comparison with FS N 1. This is because for a relatively smallFS, themean value of cu has amore significant influence on the probabil-ity of slope failure compared with the COVcu. To further explain thisfinding, Fig. 5 shows the probabilities of slope failure for various valuesof COVcu and FS using the single random variable approach. The proba-bilities of slope failure increase with COVcu for a relative high FS. WhenFS is below 0.96, however, COVcu = 0.1 leads to higher probabilities ofslope failure than those associated with COVcu exceeding 0.1.

4.2. Application to a c–ϕ slope

In the second example, the stability of a c–ϕ slope studied byGriffiths and Fenton (2004) is investigated again. Like the first example,a typical finite element model of the considered c–ϕ slope is presentedin Fig. 6. The finite element mesh is identical to that in Fig. 2, which canalso lead to almost no variance reductionwhen the squared exponentialautocorrelation model with autocorrelation distances of lh = 25 m and

1.0

FS =0.90

(lh= 25m, l

v= 2.5m) (l

h=l

v= 1000m)

FS =1.00

(lh= 25m, l

v= 2.5m) (l

h=l

v= 1000m)

FS =1.15

(lh= 25m, l

v= 2.5m) (l

h=l

v= 1000m)

FS =1.30

(lh= 25m, l

v= 2.5m) (l

h=l

v= 1000m)

FS =1.45

(lh= 25m, l

v= 2.5m) (l

h=l

v= 1000m)

FS =1.60

(lh= 25m, l

v= 2.5m) (l

h=l

v= 1000m)

ndrained shear strength and factors of safety.

0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.50.0

0.2

0.4

0.6

0.8

1.0

FS

= 0

.96

COVcu= 0.1

COVcu= 0.3

COVcu= 0.5

COVcu= 0.7

COVcu= 0.9

Pro

babi

lity

of fa

ilure

Factor of safety

Fig. 5. Effect of FS on probability of failure for various COVs of undrained shear strengthusing single random variable model.

Table 4Statistical properties of soil parameters for a c–ϕ slope.

Parameter Mean COV

Cohesion c (kPa) 10 0.3Friction angle ϕ (°) 20 0.2Young's modulus E (MPa) 100 –

Poisson's ratio ν 0.3 –

Unit weight γ (kN/m3) 20 –

126 S.-H. Jiang et al. / Engineering Geology 168 (2014) 120–128

lv = 2.5 m is adopted. Table 4 summarizes the statistical properties ofthe soil parameters for the c–ϕ slope. The cohesion and friction angleare modeled as cross-correlated lognormal random fields. Again,Young'smodulus E, Poisson's ratio v and unit weightγ are treated as de-terministic quantities. The critical failure surface obtained from the de-terministic analysis passes the slope toe, which is significantly differentfrom that for the undrained clay slope in Fig. 2. Using themean values ofthe cohesion and friction angle, the minimum factor of safety of the c–ϕslope is 1.780 from the finite element analysis incorporating an auto-matic search algorithm. Also, the coordinates of the centroids of 910 fi-nite elements are extracted and this deterministic slope stability modelfile is saved as an input file named “FEM-FS.xml”.

Unlike the first example, 20 independent standard normal randomvariables are needed to discretize cross-correlated lognormal randomfields of c and ϕ when a value of n = 10 is selected for each spatiallyvarying variable. The sample matrix, ξ, with dimensions of 20 × 462 isfirst generated for the 2nd order Hermite PCE using the Latin hypercubesampling points (Choi et al., 2004). The number of samples, 462, is usedbecause it is twice the number of coefficients, Nc, based on the regres-sion based approach (Isukapalli et al., 1998). Then, the independentstandard normal sample matrix, ξ, is transformed into the correlatedstandard normal sample matrix, χ, using Eq. (6) by the Cholesky de-composition of Rc,ϕ. The sample matrix χ20 × 462 and coordinates (xo,i,yo,i), in which i = 1, 2,⋯, 910, are used to simulate the 2-D cross-correlated lognormal random fields of c and ϕ using Eqs. (6)–(9). Twoparameter matrices of spatially variable c and ϕ in which each hasdimensions of 910 × 462 in the physical space are obtained using theKL expansion. These 910 pairs of values of c and ϕ in each column ofthe parameter matrices are assigned to each finite element accordingto the order of coordinates (xo,i, yo,i). Like the first example, 462 different

Distan

0 5 10

Ele

vatio

n (m

)

0

2

4

6

8

10

Fig. 6. FEMmodel of the homogene

new “FEM-FS.xml” input files and the corresponding factors of slopesafety, FS = (FS1, FS2,…, FS462), can be obtained. Finally, the unknowncoefficients in the 2nd order Hermite PCE are determined using the re-gression based approach. Similarly, the probability of slope failure canalso be obtained using direct MCS with 106 samples.

Table 5 shows the reliability results of a c–ϕ slope for autocorrelationdistances of lh = 25 m and lv = 2.5 m and ρc,ϕ = 0. The probabilitiesof failure obtained from the 2nd order PCEs with 462 and 2500 runs offinite element model are 6.26E-4 and 4.32E-4, respectively. This lowprobability of failure is often required in most geotechnical structuresmanagement demands and corresponds to thepractical cases. However,for the same problem, the MCS requires more than one million runs offinite element model to produce sufficiently accurate results when thecoefficient of variation of pf below 10% is satisfied. This is almost impos-sible for complex slope reliability problems. In Table 5, the resultsobtained from the 2nd order PCE are consistent with those obtainedfrom 3rd order PCE. Taking the reliability index for the 3rd order PCEas the “exact” solution, the relative error in reliability index for the2nd order PCE is below 5.0%, which indicates that the 2nd order PCEhas converged and canproduce satisfactory results based on the conver-gence property of the PCE (Mollon et al., 2011; Li et al., 2013a). For com-putational efficiency, the 2nd order PCE is also employed in thefollowing.

To account for the effect of cross-correlation between cohesion andfriction angle on the slope reliability, the cross-correlation coefficientρc,ϕ is needed. Several studies (Lumb, 1970; Wolff, 1985; Tang et al.,2012) reported values of ρc,ϕ. The range of −0.5 ≤ ρc,ϕ ≤ 0.5 is usedin this study for illustration. Fig. 7 compares the probabilities of slopefailure for the random field approach with autocorrelation distances oflh = 25 m and lv = 2.5 m and the random variable approach. Thecross-correlation between c and ϕ has a significant influence on theprobability of slope failure. For the autocorrelation distances oflh = 25 m and lv = 2.5 m, the probability of failure changes severalorders of magnitude (i.e., increasing from 3.0E-06 to 4.97E-03) whenρc,ϕ varies from−0.5 to 0.5. As expected, the probability of slope failuredecreases when the negative cross-correlation becomes stronger andincreases with positive cross-correlation coefficient. Therefore, theprobability of failure under the assumption of independence betweenc and ϕmay be severely biased if the actual cross-correlation is positiveor negative. It can also be noted that the random variable approach

ce (m)

15 20 25 30

ous c–ϕ slope with FS = 1.780.

Table 5Reliability results of a c–ϕ slope for lh = 25 m and lv = 2.5 m.

Method μFS σFS δFS κFS pf (%) β

2nd order PCE (Np = 462) 1.721 0.227 0.171 3.175 6.26E-4 3.2272nd order PCE (Np = 2500) 1.721 0.228 0.243 3.185 4.32E-4 3.3313rd order PCE (Np = 2500) 1.721 0.236 0.323 3.286 3.55E-4 3.386

127S.-H. Jiang et al. / Engineering Geology 168 (2014) 120–128

underestimates the probability of failure for large positive correlationcoefficient between cohesion and friction angle, which is consistentwith the observation in Griffiths et al. (2011).

5. Conclusions

A non-intrusive stochastic finite elementmethod has been proposedfor slope reliability analysis considering spatial variability of shearstrength parameters. The KL expansion is adopted to discretize the 2-D cross-correlated non-Gaussian random fields of spatially variableshear strength parameters. Two illustrative examples of slope reliabilityanalysis are investigated to demonstrate the capacity and validity of theproposed method. Several conclusions are drawn from this study:

(1) The proposed method does not require the user to modifyexisting deterministic finite element codes. Moreover, the deter-ministic finite element analysis and the probabilistic analysis aredecoupled. The proposed method provides a practical tool forreliability problems involving complex finite element analysis.

(2) The non-intrusive stochastic finite elementmethod can efficient-ly evaluate the slope reliability in the presence of spatial variabil-ity in shear strength parameters. It can reduce the number ofcalls to the deterministic finite element model substantially andis much more efficient than the LHS method. Additionally, thismethod can yield satisfactory results for low failure risk corre-sponding to most practical cases.

(3) Ignoring spatial variability of shear strength parameters will re-sult in unconservative estimates of the probability of slope failureif the coefficients of variation of shear strength parametersexceed a critical value or the factor of slope safety is relativelylow. The critical coefficient of variation of shear strength param-eters increases with increasing the factor of slope safety.

(4) The variation of probability of slope failure highly depends onthe factor of slope safety. The lower the value of factor of safety(i.e., FS = 0.9 in the first example), the more likely it is that theshear strength parameters with low variability will overestimatethe probability of slope failure, which is conservative for slope

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

0.000

0.001

0.002

0.003

0.004

0.005

Pro

babi

lity

of fa

ilure

Correlation coefficient

Random field approach (lh= 25m, l

v= 2.5m)

Random variable approach

Fig. 7. Effect of cross-correlation between cohesion and friction angle on probability offailure.

safety assessment.(5) When the spatial autocorrelation of the shear strength parame-

ters is very weak, more KL expansion terms are needed to pro-duce sufficiently accurate reliability results. In order to improvethe computational efficiency, other high efficient polynomialchaos expansions such as the sparse polynomial chaos expansionshould be incorporated into the non-intrusive stochastic finiteelement method.

Acknowledgments

This work was supported by the National Science Fund for Distin-guished Young Scholars (Project No. 51225903), the National BasicResearch Program of China (973 Program) (Project No. 2011CB013506)and the National Natural Science Foundation of China (Project No.51329901).

References

Al-Bittar, T., Soubra, A.-H., 2013. Bearing capacity of strip footings on spatially randomsoils using sparse polynomial chaos expansion. Int. J. Numer. Anal. MethodsGeomech. 37 (13), 2039–2060.

Ching, J., Phoon, K.K., 2013. Effect of element sizes in random field finite element simula-tions of soil shear strength. Comput. Struct. 126, 120–134.

Cho, S.E., 2007. Effects of spatial variability of soil properties on slope stability. Eng. Geol.92 (3–4), 97–109.

Cho, S.E., 2010. Probabilistic assessment of slope stability that considers the spatial variabilityof soil properties. J. Geotech. Geoenviron. 136 (7), 975–984.

Cho, S.E., 2012. Probabilistic analysis of seepage that considers the spatial variability ofpermeability for an embankment on soil foundation. Eng. Geol. 133–134, 30–39.

Choi, S.K., Canfield, R., Grandhi, R., Pettit, C., 2004. Polynomial chaos expansion with Latinhypercube sampling for estimating response variability. AIAA J. 42 (6), 1191–1198.

Der Kiureghian, A., Ke, J.-B., 1988. The stochastic finite element method in structural reli-ability. Probabilistic Eng. Mech. 3 (2), 83–91.

Duncan, J.M., 2000. Factors of safety and reliability in geotechnical engineering. J. Geotech.Geoenviron. 126 (4), 307–316.

El-Ramly, H., Morgenstern, N.R., Cruden, D.M., 2003. Probabilistic stability analysis of atailings dyke on presheared clay-shale. Can. Geotech. J. 40 (1), 192–208.

Farias, M.M., Naylor, D.J., 1998. Safety analysis using finite elements. Comput. Geotech. 22(2), 165–181.

Fenton, G.A., Griffiths, D.V., 2003. Bearing capacity prediction of spatially random c–φsoils. Can. Geotech. J. 40 (1), 54–65.

GEO-SLOPE International Ltd., 2010a. Stress–Deformation Modeling with SIGMA/W 2007Version: An Engineering Methodology [Computer Program]. GEO-SLOPE Internation-al Ltd., Calgary, Alberta, Canada.

GEO-SLOPE International Ltd., 2010b. Stability ModelingWith SLOPE/W 2007 Version: AnEngineering Methodology [Computer Program]. GEO-SLOPE International Ltd., Calgary,Alberta, Canada.

Ghanem, R.G., Spanos, P.D., 2003. Stochastic Finite Element: A Spectral Approach. Revisedversion Dover Publication, Inc., Mineola, New York.

Griffiths, D.V., Fenton, G.A., 2004. Probabilistic slope stability analysis by finite elements.J. Geotech. Geoenviron. 130 (5), 507–518.

Griffiths, D.V., Huang, J.S., Fenton, G.A., 2011. Probabilistic infinite slope analysis. Comput.Geotech. 38 (4), 577–584.

Huang, S.P., 2001. Simulation of Random Processes Using Karhunen–Loeve Expansion.(Ph.D. thesis) National University of Singapore, Singapore.

Isukapalli, S.S., Roy, A., Georgopoulos, P.G., 1998. Stochastic response surface methods foruncertainty propagation: application to environmental and biological systems. RiskAnal. 18 (3), 351–363.

Ji, J., Liao, H.J., Low, B.K., 2012. Modeling 2-D spatial variation in slope reliability analysisusing interpolated autocorrelations. Comput. Geotech. 40, 135–146.

Laloy, E., Rogiers, B., Vrugt, J.A., Mallants, D., Jacques, D., 2013. Efficient posterior explora-tion of a high-dimensional groundwater model from two-stage MCMC simulationand polynomial chaos expansion. Water Resour. Res. 49 (5), 2664–2682.

Li, H., Zhang, D., 2007. Probabilistic collocationmethod for flow in porousmedia: compar-isons with other stochastic method. Water Resour. Res. 43 (W09409), 44–48.

Li, D.Q., Chen, Y.F., Lu, W.B., Zhou, C.B., 2011. Stochastic response surface method forreliability analysis of rock slopes involving correlated non-normal variables. Comput.Geotech. 38 (1), 58–68.

Li, D.Q., Jiang, S.H., Chen, Y.G., Zhou, C.B., 2013a. A comparative study of three collocationpoint methods for odd order stochastic response surface method. Struct. Eng. Mech.45 (5), 595–611.

Li, D.Q., Tang, X.S., Phoon, K.K., Chen, Y.F., Zhou, C.B., 2013b. Bivariate simulation usingcopula and its application to probabilistic pile settlement analysis. Int. J. Numer.Anal. Methods Geomech. 37 (6), 597–617.

Li, D.Q., Qi, X.H., Phoon, K.K., Zhang, L.M., Zhou, C.B., 2013c. Effect of spatially variableshear strength parameters with linearly increasing mean trend on reliability of infi-nite slopes. Struct. Saf.. http://dx.doi.org/10.1016/j.strusafe.2013.08.005.

Liu, W.K., Belytschko, T., Mani, A., 1986. Random field finite elements. Int. J. Numer.Methods Eng. 23 (10), 1831–1845.

128 S.-H. Jiang et al. / Engineering Geology 168 (2014) 120–128

Low, B.K., Lacasse, S., Nadim, F., 2007. Slope reliability analysis accounting for spatialvariation. Georisk 1 (4), 177–189.

Lumb, P., 1970. Safety factors and the probability distribution of soil strength. Can.Geotech. J. 7 (3), 225–242.

Mollon, G., Dias, D., Soubra, A.-H., 2011. Probabilistic analysis of pressurized tunnelsagainst face stability using collocation-based stochastic response surface method.J. Geotech. Geoenviron. 137 (4), 385–397.

Phoon, K.K., Kulhawy, F.H., 1999. Characterization of geotechnical variability. Can.Geotech. J. 36 (4), 612–624.

Phoon, K.K., Huang, S.P., Quek, S.T., 2002. Implementation of Karhunen–Loeve expansion forsimulation using a wavelet-Galerkin scheme. Probabilistic Eng. Mech. 17 (3), 293–303.

Srivastava, A., Sivakumar Babu, G.L., 2009. Effect of soil variability on the bearing capacityof clay and in slope stability problems. Eng. Geol. 108 (1–2), 142–152.

Srivastava, A., Sivakumar Babu, G.L., Haldar, S., 2010. Influence of spatial variability ofpermeability property on steady state seepage flow and slope stability analysis.Eng. Geol. 110 (3–4), 93–101.

Stefanou, G., 2009. The stochastic finite element method: past, present and future.Comput. Methods Appl. Mech. Eng. 198 (9–12), 1031–1051.

Tabarroki, M., Ahmad, F., Banaki, R., Jha, S., Ching, J., 2013. Determining the factors of safe-ty of spatially variable slopes modeled by random fields. J. Geotech. Geoenviron. 139(12), 2082–2095.

Tang, X.S., Li, D.Q., Chen, Y.F., Zhou, C.B., Zhang, L.M., 2012. Improved knowledge-basedclustered partitioning approach and its application to slope reliability analysis.Comput. Geotech. 45, 34–43.

Tang, X.S., Li, D.Q., Rong, G., Phoon, K.K., Zhou, C.B., 2013. Impact of copula selection ongeotechnical reliability under incomplete probability information. Comput. Geotech.49, 264–278.

Vanmarcke, E.H., 1977. Probabilistic modeling of soil profiles. J. Geotech. Eng. Div. 103(11), 1227–1246.

Vanmarcke, E.H., 2010. Random Fields: Analysis and Synthesis. Revised and ExpandedNew Edition. World Scientific Publishing, Beijing.

Vořechovský, M., 2008. Simulation of simply cross-correlated random fields by seriesexpansion methods. Struct. Saf. 30 (4), 337–363.

Wang, Y., Cao, Z.J., Au, S.K., 2011. Practical reliability analysis of slope stability byadvanced Monte Carlo simulations in a spreadsheet. Can. Geotech. J. 48 (1),162–172.

Wolff, T.F., 1985. Analysis and Design of Embankment Dam Slopes: A Probabilistic Ap-proach. (Ph. D. thesis) Purdue University, Lafayette, Ind, USA.

Zhang, J., Huang, H.W., Juang, C.H., Li, D.Q., 2013. Extension of Hassan and Wolff methodfor system reliability analysis of soil slopes. Eng. Geol. 160, 81–88.

Zhu, H., Zhang, L.M., 2013. Characterizing geotechnical anisotropic spatial variations usingrandom field theory. Can. Geotech. J. 50 (7), 723–734.